[EM] Wasted votes in party list, and cloneproofing

[Kristofer Munsterhjelm]

Suppose we have a party list method, and for simplicity's sake suppose 
it's quota based rather than divisor based. Also suppose part of the 
result goes like this:

Party L1: 3.3 seats
Party L2: 2.3 seats
Party L3: 1.3 seats
Party L4: 0.1 seats

and that the largest remainders are distributed so that all these L 
parties get their seat shares rounded down. It then seems like there are 
some wasted votes. Suppose the voters compromised instead, voting:

Party L1: 4 seats
Party L2: 2 seats
Party L3: 1 seats
Party L4: 0 seats

Now the L bloc has got one more seat by their voters compromising so 
that the wasted shares end up giving L1 another seat.

I'd like to have a criterion that somewhat formalizes whether a method 
is vulnerable to that kind of compromising incentive, since the 
scenarios could potentially get very complex. Suppose the L-voters 
compromise like this, then some other bloc's voters (say R-voters) 
compromise in return; would that render the L-voters' strategy

But then, after thinking for a bit, I thought that the independence of 
clones criterion would work. Suppose that the L-bloc all vote the 
L-parties ahead of everybody else, i.e.

x_0: L1>L2>L3>L4>[others]
x_2: L1>L2>L4>L3>[others]
...
x_24: L4>L3>L2>L1>[others]

and that similarly, all voters who rank one of the L parties at position 
p, rank all of them before ranking someone else,

then we could define a party list independence of clones criterion:

"Cloning a party should not affect the number of seats given to the 
other parties, assuming that every party fields at least as many 
candidates as there are seats". By pigeonhole, if any of the clones lose 
or gain seats compared to the pre-clone party, those seats will be 
acquired or lost by some party not part of the cloning process. The 
second clause prevents false positives from the parties having too few 
candidates to claim all their seats.

And it seems that if a method is cloneproof, the problem above can't 
happen, as long as every voter rank the parties of the bloc next to each 
other.

Suppose we have

Party L1: 3.3 seats -> 3 seats
Party L2: 2.3 seats -> 2 seats
Party L3: 1.3 seats -> 1 seat
Party L4: 0.1 seats -> 0 seats

Now, declone the L bloc to a single L party. Either L will have 6 seats 
or it won't. If it doesn't, then we have a clone failure when we run the 
scenario in reverse: cloning L into {L1, L2, L3, L4} made the number of 
seats change. Call that failure "failure type 1".

So suppose it didn't change. Then we can clone L into {L1, L2, L3, L4} 
so that

Party L1: 4 seats
Party L2: 2 seats
Party L3: 1 seats
Party L4: 0 seats

happens, which means that cloning paid off in that L got an additional 
seat. Call that "failure type 2".

So if it's possible to redistribute the votes to get more seats, the 
method will fail the independence of clones criterion - *if every voter 
ranks the L-parties adjacent to each other*. Either the method will 
exhibit failure type 1, type 2, or both.

This doesn't make the method immune to compromise just like the 
independence from clones criterion doesn't make single-winner methods 
immune to compromise; but at least a significant part of a "wasted bloc" 
effect will be detectable by independence of clones failure.

Clearly, Plurality style party list (Webster, etc) fails the 
independence of clones criterion. Let there be two parties (A and B) 
with a 50-50 split. Then clone B into as many parties as there are 
voters, and each voter votes for one of them. Unless there are very many 
seats, A will get all of them. Or just consider that the one-seat 
instance becomes Plurality, and Plurality is well known for clone problems.

Similarly, any procedure where you run a method to get scores for each 
party (and this method doesn't take the number of seats as input), and 
you then run a divisor method on the scores to apportion seats, must 
fail. This because the base method doesn't know how many seats there are 
and so can't quantize properly. In essence, virtual voters cast 
Plurality-style ballots based on the first method so that the sum of 
votes for each party is that party's score. Since manipulating the 
Plurality-style ballots can be beneficial, so can manipulating the score 
be, and the second method can't counter this because it doesn't know 
about anything other than the scores provided by the first method (and 
the number of seats).